Not only eastern cultures have the concept.
In German there is the phrase 'a steady water drop hollows a stone' and it actually traces back to the Roman author Ovid 'gutta cavat lapidem'.
Perseverance is just so much harder than instant gratification.
Well you can't really reject all elements of American culture that have their roots in Europe as not American or else you end up without a whole lot to work with.
I always thought story-myths like Star Wars and The Karate Kid fulfilled the role of fairytales and parables for the comparatively young culture of United States. I'm only half-kidding. I bet Yoda has a few choice quotes related to exactly this subject?
It seems to me that that's lost the idea of incremental improvement. It's about taking the time to do things right, but there's no sense of making it better.
Hm, that's a good point. A favorite saying around here is, "If you don't have time to do it right, you don't have time to do it over!" There's a big emphasis on doing something right, and if you have to go back and improve it later, that's a failure.